Question

Find the angle of inclination of the line represented by the equation. $$ x+y-8=0 $$

Short answer

The angle of inclination for the line represented by the equation \(x+y-8=0\) can be found by first determining the slope of the line using the slope-intercept form, which is \(m=-1\). Using the formula \(tan(\theta) = m\), we find that the angle of inclination can be either \(\theta = 135^\circ\) or \(\theta = 315^\circ\).

Step-by-step solution

Convert the given equation into the slope-intercept form

To convert the equation \(x+y-8=0\) into slope-intercept form, we must isolate \(y\) on one side of the equation. Subtracting \(x\) from both sides of the equation, we get: \(y = -x + 8\) Now, the equation is in the slope-intercept form \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept.

Identify the slope of the line

From the slope-intercept equation obtained in Step 1, \(y = -x + 8\), we can see that the slope of the line is \(m=-1\).

Find the angle of inclination using the formula

We'll now use the formula \(tan(\theta) = m\) to find the angle of inclination. We have the slope \(m = -1\), so the equation becomes: \(tan(\theta) = -1\) Now, we'll find the angle \(\theta\) by taking the inverse tangent (arctan) of both sides: \(\theta = arctan(-1)\) Since the tangent function is negative in the second and fourth quadrants, we can conclude that the angle of inclination lies in either the second or the fourth quadrant. From this, we get: \(\theta = 135^\circ \text{ or } \theta = 315^\circ\) (in degrees)